Real Quantum Theory: Foundations & Distinctions

• Real Quantum Theory is a formulation of quantum mechanics over real Hilbert spaces, where all states, observables, and transformations are represented by real-valued matrices.
• It exhibits distinctive operational features such as the failure of local tomography and modified entanglement measures compared to standard complex quantum mechanics.
• Experimental network tests and symplectic embedding methods highlight quantitative gaps between RQT and CQT, guiding advances in foundational research.

Real Quantum Theory (RQT) is a variant of quantum mechanics built on real Hilbert spaces rather than the standard complex Hilbert space formalism. In RQT, all states, transformations, and observables are represented by real-valued matrices and vectors, and all inner products, superposition principles, and other linear-algebraic operations are conducted over the field of real numbers, $\mathbb{R}$, rather than $\mathbb{C}$. The resulting theory shares most structural features with standard quantum theory but differs crucially in its representation of state spaces, composite systems, and the physical interpretation of phases, leading to both operational equivalences and distinctive foundational subtleties.

1. Mathematical Foundations of Real Quantum Theory

In RQT, every quantum system is modeled by a real Hilbert space ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$ equipped with the usual real inner product. Pure states correspond to unit vectors $\ket{\psi}\in\mathbb{R}^n$ with $\|\psi\|=1$, and mixed states are real symmetric positive semidefinite matrices: $\rho\in\mathbb{R}^{n\times n}$, $\rho^T = \rho$, $\rho \geq 0$, $\Tr\rho = 1$. Observables are real symmetric matrices $O\in\mathbb{R}^{n\times n}$, and their spectral decompositions mirror those in the complex case. The Born rule for measurement statistics carries over: the probability of outcome $\mathbb{C}$0 for observable $\mathbb{C}$1 in state $\mathbb{C}$2 is $\mathbb{C}$3, where $\mathbb{C}$4 are real symmetric projectors.

For dynamics, reversible transformations are given by real-orthogonal operators. General physical transformations are real-matrix CPTP (completely positive trace-preserving) maps. Composition of systems is defined using the real tensor product, so a composite of $\mathbb{C}$5 and $\mathbb{C}$6 is modeled by $\mathbb{C}$7, with observables and states respecting this real structure (Finkelstein, 2021, Yīng et al., 9 Jun 2025).

2. Composition, Local Tomography, and Operational Structure

A critical structural distinction between RQT and complex quantum theory (CQT) concerns the phenomenon of local tomography. In CQT, the global state of a composite system can be fully reconstructed by measuring local observables—this is local discriminability. In RQT, generic global states of $\mathbb{C}$8 contain additional parameters (“hidden” degrees of freedom) that are invisible to product measurements: local tomography fails except for special cases. As a result, the structure of entanglement and the behavior of independent sources diverge between RQT and CQT.

This lack of local tomography in RQT leads to operationally equivalent states that are distinct mathematically but indistinguishable via standard measurements (Hoffreumon et al., 19 Mar 2026, Belenchia et al., 2012). This property has direct consequences for experimental tests and the foundation of quantum information protocols.

3. Embeddings, Tensor Product Rules, and Universality Rebuttals

A classical argument (Stueckelberg, extended by Finkelstein (Finkelstein, 2021)) shows that any complex quantum theory can be embedded within a real Hilbert space of doubled dimension by coupling to an ancillary “phase” rebit. For pure or mixed states, this embedding takes the form: $\mathbb{C}$9 where ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$0 are projectors onto ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$1. Observables and channels are similarly embedded. This real embedding preserves all measurable outcome probabilities, making RQT, when formulated with nonstandard tensor products or appropriate embeddings, operationally indistinguishable from ordinary quantum mechanics for all practical purposes—including Bell tests, state/process tomography, and computation (Finkelstein, 2021, Hoffreumon et al., 19 Mar 2026).

However, if one demands that RQT respects strict “standard” tensor product structure (Kronecker product without ancillary extension), there arise scenarios—especially in network nonlocality experiments—where RQT and CQT part ways.

4. Experimental Separation, Network Bell Inequalities, and Scalability

Recently, fundamental differences between RQT and CQT were experimentally demonstrated in multipartite network scenarios. In the entanglement-swapping network of Renou et al., it was shown theoretically and experimentally that CQT can attain violations of a multipartite Bell-type functional (the CHSH${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$2 or more general witnesses) that RQT cannot reach under real-only, standard tensor-product assumptions (Renou et al., 2021, Chen et al., 2021, Li et al., 2021). For example, in a three-party network, the maximal value for a certain witness is ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$3 in CQT, but only ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$4 in RQT via SDP bounds. These CQT-predicted correlations have been observed in both superconducting-qubit (Chen et al., 2021) and photonic (Li et al., 2021) experiments, exceeding the RQT bound by many standard deviations (e.g., 43${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$5).

The gap between RQT and CQT predictions can be amplified in star networks with ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$6 parties: for the appropriate Bell-type functional, the CQT:RQT ratio grows linearly in ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$7, becoming arbitrarily large as the network size increases (Sarkar et al., 12 Mar 2025). No fixed real-Hilbert-space model respecting standard tensor products can replicate these correlations in large networks.

A key point is that these results only constrain RQT formulated with the naive tensor-product structure. Alternative RQT formulations with modified composition rules or embedding schemes remain unfalsified by such experiments (Hoffreumon et al., 19 Mar 2026, Maioli et al., 21 Apr 2026).

5. Distinguishability, Reference Frames, and Symmetric Foil Theories

RQT arises as the “swirled” symmetry subtheory of complex quantum mechanics, i.e., the set of states, observables, and channels invariant under collective antiunitary (time-reversal) conjugation (Yīng et al., 9 Jun 2025). This stands in contrast to “twirled” (unitary-symmetry) subtheories, which cannot be distinguished from CQT by any experiment given only causal structure and no additional assumptions (the reference-frame simulation argument). By contrast, the absence of a shared complex phase (reference frame) among participants indeed constitutes a resource distinguishing CQT from RQT, and is precisely what is tested by bilocality and network nonlocality experiments.

In scenarios where the causal structure (the pattern of allowed interactions and independence) supports it, RQT and CQT can yield experimentally distinct predictions, but only if one insists on strict locality and independence for sources and measurement processes (Yīng et al., 9 Jun 2025, Renou et al., 2021).

6. Generalizations: Causal Order, Process Matrices, and Reverse Hierarchies

Within the process-matrix framework for indefinite causal order, the RQT–CQT hierarchy reverses: with definite causal order, CQT can generate correlations not allowed by RQT, but under indefinite causal order, RQT enables a strictly larger set of process correlations (Surace et al., 28 May 2026). Finite-unitary twirled theories add no new process correlations, but the antiunitary symmetry of RQT allows the realization of process matrices violating causal inequalities beyond complex QT limits. This points to a complex interplay between the algebraic structure of the field and the nature of causal order in multipartite scenarios.

7. Real-Kähler Space, Deeper Geometric Structure, and Conceptual Implications

A rigorous formulation of quantum mechanics over real numbers is furnished by “Kähler space” constructions, providing a real vector space ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$8 with compatible metric ${\mathcal H}_\mathbb{R}\cong\mathbb{R}^n$9, symplectic form $\ket{\psi}\in\mathbb{R}^n$0, and complex structure $\ket{\psi}\in\mathbb{R}^n$1 ($\ket{\psi}\in\mathbb{R}^n$2). The complex Hilbert space is then simply a compact encoding of this richer real-geometric data ($\ket{\psi}\in\mathbb{R}^n$3 as shorthand for $\ket{\psi}\in\mathbb{R}^n$4) (Maioli et al., 21 Apr 2026). Composite systems are equipped with a symplectic tensor product $\ket{\psi}\in\mathbb{R}^n$5 replacing the naive Kronecker product, yielding an exact monoidal equivalence with CQT—reproducing all quantum predictions, including the maximum possible Bell inequality violations. The imaginary unit $\ket{\psi}\in\mathbb{R}^n$6 appears as $\ket{\psi}\in\mathbb{R}^n$7, and phase dynamics as real rotations under $\ket{\psi}\in\mathbb{R}^n$8.

This real-geometric perspective resolves historical debates about the “necessity” of complex numbers: complex amplitudes concisely encode physical rotations and interference arising from real, compatible structures. Any experimental “gap” arises only when the full geometric data of the system—including the symplectic composition—are not supplied (Maioli et al., 21 Apr 2026).

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References:

• “In defense of real quantum theory” (Finkelstein, 2021)
• “Quantum theory based on real numbers can be experimentally falsified” (Renou et al., 2021)
• “Ruling out real-valued standard formalism of quantum theory” (Chen et al., 2021)
• “Testing real quantum theory in an optical quantum network” (Li et al., 2021)
• “Gap between quantum theory based on real and complex numbers is arbitrarily large” (Sarkar et al., 12 Mar 2025)
• “Quantum theory based on real numbers cannot be experimentally falsified” (Hoffreumon et al., 19 Mar 2026)
• “On whether quantum theory needs complex numbers: the foil theories perspective” (Yīng et al., 9 Jun 2025)
• “Quantum mechanics over real numbers fully reproduces standard quantum theory” (Maioli et al., 21 Apr 2026)
• “Indefinite Causal Order Reverses the Real-Complex Hierarchy” (Surace et al., 28 May 2026)
• “Universality of Computation in Real Quantum Theory” (Belenchia et al., 2012)
• “Optimal Information Transfer and Real-Vector-Space Quantum Theory” (Wootters, 2013)